Mod-two cohomology ring of real projective space ($H^{\ast }({\mathbb{RP}}^m;{\mathbb{F}}_2)$)

The mod-two cohomology ring of real projective space is

The unit sphere bundle of the tautological line is the double cover $S^n\to {\mathbb{RP}}^n$. In its Gysin sequence, multiplication by the mod-two Euler class $a$ is an isomorphism in successive interior degrees, since the sphere has no interior cohomology. Connectedness in degree zero makes multiplication into degree one injective. At the top, vanishing above dimension $n$ makes the transfer from $H^n(S^n;{\mathbb{F}}_2)$ surjective; together with that injection it makes the top group one-dimensional. Thus every $a^j$, $0\le j\le n$, is the nonzero generator and the next power vanishes.

For any continuous map between the displayed Real projective spaces, the induced map on cohomology sends the degree-one generator to $\alpha x+\beta y$. Its $2n$th power is $\left(\genfrac{}{}{0ex}{}{2n}{n}\right){\alpha }^n{\beta }^n{x}^n{y}^n$, since all other terms exceed the truncation exponent of one factor. The coefficient is zero modulo two because $\left(\genfrac{}{}{0ex}{}{2n}{n}\right)=2\left(\genfrac{}{}{0ex}{}{2n-1}{n-1}\right)$. Naturality of the evaluation pairing then makes the top induced map on homology zero. This obstruction does not apply to $n=0$, when both spaces are points.